Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T19:30:54.739Z Has data issue: false hasContentIssue false
  • English
  • Français

On theories categorical in their own power

Published online by Cambridge University Press:  12 March 2014

H. Jerome Keisler
H. Jerome Keisler*
Affiliation:
University of Wisconsin, Madison, Wisconsin
Get access Rights & Permissions [Opens in a new window]

Extract

A theory T is said to be categorical in power κ iff T has a model of power κ and any two models of power κ are isomorphic.

It was conjectured by Morley [4] that if T is a theory in a language with κ > ω symbols and T is categorical in power κ, then T has a model of power < κ. The aim of this paper is to prove the following theorem.

Theorem A. Let κ be a regular cardinal such that ω < κ < 2ω. Let T be a theory in a language with κ symbols such that T is categorical in power κ. Then:

(a) T has a model of power < κ.

(b) T is categorical in all powers μ ≥ κ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 240 - 244
Copyright
Copyright © Association for Symbolic Logic 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chang, C. C., On the formula “there exists x such that f(x) for all f ∃ F”, Notices of the American Mathematical Society, vol. 11 (1964), p. 587 (Abstract).Google Scholar
[2]Chang, C. C. and Keisler, H. J., Model theory, Appleton-Century-Crofts, New York, 1971.Google Scholar
[3]Kreisel, G. and Krivine, J. L., Elements of mathematical logic, North-Holland, Amsterdam, 1967.Google Scholar
[4]Morley, M., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.CrossRefGoogle Scholar
[5]Ressayre, J. P., Sur les theories duprimier ordre categorique en un cardinal, mimeographed, Paris, France.Google Scholar
[6]Rowbottom, F., The Los conjecture for uncountable theories, Notices of the American Mathematical Society, vol. 11 (1964), p. 248 (Abstract).Google Scholar
[7]Shelah, S., On theories T categorical in ∣T∣, this Journal, vol. 35 (1970), pp. 73–82.Google Scholar
[8]Shelah, S., On stable theories, Israel journal of mathematics (to appear).Google Scholar
[9]Vaught, R., Denumerable models of complete theories, Infinitistic methods (Symposium on Foundations of Mathematics at Warsaw, 1959), Pergamon Press, Warsaw, 1961, pp. 303–321.Google Scholar
[10]Makinson, D. C., On the number of ultrafilters of an infinite Boolean algebra, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 121–122.CrossRefGoogle Scholar